Parametrization of local CR automorphisms by finite jets and applications Academic Article

For any real-analytic hypersurface M C N Msubset mathbb {C}^N , which does not contain any complex-analytic subvariety of positive dimension, we show that for every point p M pin M the local real-analytic CR automorphisms of M M fixing p p can be parametrized real-analytically by their p ell _p jets at p p . As a direct application, we derive a Lie group structure for the topological group Aut ( M , p ) operatorname {Aut}(M,p) . Furthermore, we also show that the order p ell _p of the jet space in which the group Aut ( M , p ) operatorname {Aut}(M,p) embeds can be chosen to depend upper-semicontinuously on p p . As a first consequence, it follows that given any compact real-analytic hypersurface M M in C N mathbb {C}^N , there exists an integer k k depending only on M M such that for every point